Black Holes are regions of space in which gravitational fields are so
strong that no particle or signal can escape the pull of gravity.
The boundary of this no-escape region is called the event horizon, since
distant observers outside the black hole cannot see (cannot get light from)
events inside.

Although the fundamental possibility of such an object exists within
Newton's classical theory of gravitation, Einstein's theory of gravity
makes black holes inevitable under some circumstances. Prior to the
early 1960s, black holes seemed to be only an interesting theoretical
concept with no astrophysical plausibility, but with the discovery of
quasars in 1963 it became clear that very exotic astrophysical objects
could exist. Nowadays it is taken for granted that black holes do
exist in at least two different forms. Stellar mass black holes are
the endpoint of the death of some stars, and supermassive black holes
are the result of coalescences in the centers of most galaxies,
including our own.

No signal can propagate from inside a black hole, but the
gravitational influence of a black hole is always present. (This
influence does not propagate out of the hole; it is permanently
present outside, and depends only on the total amount of mass, angular momentum,
and electric charge that have gone into forming the hole.) Black holes can be detected through the influence of
this strong gravity on the surroundings just outside the hole. In this
way, stellar mass holes produce detectable X-rays, supermassive black
holes produce a wide spectrum of electromagnetic signals, and both
types can be inferred from the orbital motion of luminous stars and matter around
them. Phenomena involving black holes of any mass can produce strong
gravitational waves, and are of interest as sources for present and
future gravitational wave detectors.

Contents

Classical vs. relativistic black holes

Something like a black hole exists within Newton's classical
theory of gravity. In that theory, an energy argument tells us that
there is an escape velocity \(v_{\rm esc}
=\sqrt{2GM/R}\)
from the surface of any spherical object of mass \(M\) and radius
\(R\ .\) If this velocity is greater than the speed of light \(c\) then light
from this object cannot escape to infinity. Thus the condition for such an "unseeable"
object is

\[R<2GM/c^2\ .\]

In the classical theory, a particle could overcome this gravity with
strong enough engines to provide the energy needed for escape. This
is not so in general relativity, Einstein's theory of gravitation. In that
theory, escaping the black hole is equivalent to
moving faster than light, an impossibility in relativity.

To understand the relativistic black hole it is useful to think of
space being dragged inward towards a gravitational center, at a faster
rate near the center than far from it. The distance at which space is
moving inward at the speed of light represents the location of the
event horizon, since no signal can progress outward through space
faster than \(c\ .\) This comparison is more than a metaphor; black
hole analog experiments with accelerating gas flows and other
phenomena are being designed.

An important difference from Newton's theory is that Einstein's,
and other relativistic theories of gravitation are nonlinear in the
sense that gravitation (as well as mass) can be a source of gravity.
Thus when a massive object collapses small enough, the tendency to
continue the collapse and form a black hole can become
unstoppable.

Stationary black holes

In the Newtonian theory, gravity is described by the potential
\(\Phi\ .\) Inside a spherical object the form of \(\Phi(r)\) depends on the
interior structure, but in the vacuum outside matter the potential
\(-GM/r\) depends only on the interior mass. Similarly, in Einstein's
theory the stationary (time-independent) spherically symmetric
exterior solution, called the Schwarzschild spacetime, depends only on
the mass of the interior object. If the interior object is small
enough, then the Schwarzschild exterior extends to small enough radius
that there is a horizon, a surface across which light cannot move
outward. This horizon radius \(R_H =2GM/c^2\) is, coincidentally, the
same as the critical radius for "unseeable" objects in Newton's
theory. (The meaning of "radius" as distance to the center is not
straightforward for the Schwarzschild solution. Radius \(R_H \)here
actually means that the area of the event horizon is \(4\pi R_H ^2\ .\))

In Einstein's theory, the "exterior" solution can be taken to apply
with no interior solution. In this case it is gravity itself, rather
than matter, that acts as the source of gravity. The inward-extended
exterior solution does not reach a center, but rather is connected via
a spacetime bridge to another universe, or another section of our own.
For an astrophysical black hole, formed from the collapse of matter, a
physical solution for the matter distribution replaces the pure vacuum
Schwarzschild solution in the interior of the black hole. This
physical solution lacks the spacetime bridge of ideal mathematical
black holes, but contains a central "singularity" where matter is
compressed to infinite density. Very close to this singularity it is
expected that the laws of general relativity will no longer apply, and
as-yet unknown laws of quantum gravity are needed.

A more general stationary black hole solution of Einstein's theory is
the Kerr solution, a vacuum spacetime with both mass and angular
momentum, and taken to represent a rotating black hole. In
its pure mathematical form the Kerr hole contains a spacetime bridge, but as in
the case of the Schwarzschild black hole this bridge is absent in
realistic black holes that form by the collapse of matter.

Unlike the Schwarzschild spacetime, the Kerr solution is not
the exterior spacetime of a material object with angular momentum. (In
fact no realistic solution has been found to join a Kerr exterior to a
material interior.) The Kerr solution only becomes the exterior
spacetime asymptotically at very late times after the collapse of an
object.

Two other exact mathematical black hole solutions are the
Reissner-Nordström spacetime, representing a hole with mass and
electrical charge, and the Kerr-Newman spacetime, representing a hole
with mass, electrical charge, and angular momentum. These spacetimes
are not astrophysically relevant, since astrophysical bodies have
negligible net electrical charge. [For a detailed description of these spacetimes
see, e.g., Misner et al. (1973), Part VII; or Wald (1984), Chap. 12.]

All of these spacetimes, including the ones with angular momentum, are
stationary: that is, they are independent of time. But in relativity
there is no unique meaning to time, so an important question is: "Just
what 'time' is it of which the stationary black holes are
independent?" The answer lies in the fact that one can assign every
spacetime point four coordinates, four labels that uniquely identify
the location of each point. One of these coordinates is called the
"coordinate time." Spacetimes that are said to be stationary, like
the spacetime of a Kerr hole, have a special property: the time
coordinate may be chosen so that the spacetime geometry is the same at
any moment of this time coordinate.

Figure 2: Coordinate time vs. proper time for the radial fall of a particle towards a Schwarzschild black hole

At large distances from a stationary black hole, where spacetime
curvatures are weak, this stationary time coordinate can be chosen
also to have another important property: to agree with the
"proper time," or ordinary clock time, of an observer at rest with
respect to the hole. Since we ourselves are more-or-less at rest (or
are at nonrelativistic velocities) very far from black holes, this
kind of stationary coordinate time is the time used in astronomical
observations. For observers near the hole, however, proper time and
stationary coordinate time will not agree. An interval of proper time
between two events is shorter (near the horizon, much shorter) than
the interval of stationary coordinate time between those two events.
The relationship between stationary coordinate and proper time is
further complicated by special relativistic time dilation for
observers who are rapidly moving. Figure 2 illustrates this by showing
the coordinate time vs. radius for a particle falling into a black
hole, and comparing it with the proper time measured by an observer
riding along with the infalling particle. The progress as measured
with proper time is in no way special at the horizon: falling
observers will not notice anything strange as they pass the point of
no return. But as described in coordinate time, the observer (and
likewise, the surface of a collapsing star) takes an infinite amount
of time to reach the horizon. Since coordinate time is the proper time
of distant observers, astronomers will see the particle reach the
horizon only in the infinite future.

The two (or more) types of time are sometimes a source of confusion in
discussions of black hole phenomena, since they often give totally
different answers to the question "how long does it take?"

Black hole parameters

Astrophysical black holes are characterized by two parameters:
their mass and their angular momentum (or spin). The
mass parameter \(M\) is equivalent to a characteristic length
\(GM/c^2=1.48\,{\rm km}(M/M_{\rm o})\ ,\) or a characteristic timescale
\(GM/c^3=4.93\times10^{-6} \,{\rm sec}(M/M_{\rm o})\ ,\) where \(M_{\rm o}\)
denotes the mass of the Sun. These scales, for
example, give the order of magnitude of the radii and periods of
near-hole orbits. The timescale also applies to the process in which a
developing horizon settles into its asymptotically stationary
form. For a stellar mass hole this is of order \(10^{-5} \,{\rm sec} \ ,\)
while for a supermassive hole of \(10^8M_{\rm o}\ ,\) it is thousands of
seconds.

For Schwarzschild holes, and approximately for Kerr holes, the horizon
is at radius \(R_H= 2GM/c^2 \ .\) At the horizon, the "acceleration of
gravity" has no meaning, since a falling observer cannot stop at the
horizon to be weighed. What is relevant at the horizon is the tidal
stresses that stretch and distort the falling observer. This tidal
stretching is given by the same expression, the gradient of the
gravitational acceleration, as in Newtonian theory\[2GM/R_H^3=c^6/(4G^2M^2)\ .\] In the case of a solar mass black hole the
tidal stress (acceleration per unit length) is enormous at the
horizon, on the order of \(3\times10^9(M_{\rm o}/M)^2\mathrm{s}^{-2}\ :\)
that is, a person would experience a
differential gravitational field of about \(10^9\) Earth gravities,
enough to rip apart ordinary materials. For a supermassive hole, by
contrast, the tidal force at the horizon is smaller by a typical
factor \(10^{10\mbox{--}16}\) and would be easily survivable. However,
at the central singularity, deep inside the event horizon, the tidal
stress is infinite.

In addition to its mass \(M\ ,\)
the Kerr spacetime is described with a spin parameter \(a\) defined
by the dimensionless expression

\[\frac{a}{M}=\frac{cJ}{GM^2}\,,\]

where \(J\) is the angular momentum of the hole.
For the Sun (based on surface rotation) this number is about 0.2, and is much
larger for many stars. Since
angular momentum is ubiquitous in astrophysics, and since it is expected
to be approximately conserved during collapse and black hole formation,
astrophysical holes are expected to have significant values of \(a/M\ ,\)
from several tenths up to and approaching unity.

The value of \(a/M\) can be unity (an "extreme" Kerr hole), but it
cannot be greater than unity. In the mathematics of general
relativity, exceeding this limit replaces the event horizon with an
inner boundary on the spacetime where tidal forces become infinite.
Because this singularity is "visible" to observers, rather than
hidden behind a horizon, as in a black hole, it is called a naked
singularity. Toy models and heuristic arguments suggest that as
\(a/M\) approaches unity it becomes more and more difficult to add
angular momentum. The conjecture that such mechanisms will always keep
\(a/M\) below unity is called cosmic censorship.

The inclusion of angular momentum changes details of the description
of the horizon, so that, for example, the horizon area
becomes

This modification of the Schwarzschild (\(a=0 \)) result is
not significant until \(a/M\) becomes very close to
unity. For this reason, good estimates can be made in many
astrophysical scenarios with \(a \) ignored.

Dynamical black holes

The event horizon is defined as the outer boundary of the region from
which there is no escape. For stationary black holes this surface is
at a fixed location in space, but more generally the horizon is
dynamical; it can grow, change shape, oscillate. In particular,
a horizon can be born.

The birth of a horizon is a change first studied by Oppenheimer and
Snyder for the collapse of spherical pressureless fluid. The general
scenario is for a horizon to be born deep inside the collapsing matter
and to spread outward.

Small changes in a horizon can be treated as perturbations, greatly
simplifying the mathematics of Einstein's equations. In this way, it
has been found that black holes have characteristic patterns of
oscillations, called quasinormal modes. These modes are like
mechanical resonances, but are highly damped by the emission of
gravitational waves, and have both periods and damping times on the
order of the characteristic time \(GM/c^3\ .\)

For large nonspherical changes in a dynamical horizon, only
supercomputer simulations can give quantitative answers. The focus of
such work has been the merger of two black holes in binary orbit
around each other, a scenario of special interest as a source of strong
gravitational wave emission. Only in 2005 were technical problems
first overcome [Pretorius (2005) (See Fig. 3.), Campanelli et al. (2006), Baker et al. (2006)] so that
an accurate picture could emerge of how two
horizons join to become a single final horizon.

There is one change that a horizon cannot make according to classical
(nonquantum) general relativity: it cannot decrease its area. But
considerations by Stephen Hawking and others of quantum effects
in black hole spacetimes suggest that radiation arising in
the close exterior of the black hole can carry off energy, and
decreases the mass (and hence horizon area) of the hole. Although no
quantum theory of relativistic gravitation currently exists, it is
generally accepted that this Hawking radiation will be a feature of
any such theory. The radiation behaves as if the horizon were a
blackbody (perfect thermal emitter) at a temperature
\(6\times10^{-8}(M_{\rm o}/M)\)K. Thus for astrophysical black holes, mass
loss by Hawking radiation is much less important than the
mass increase due to absorption of the 3K cosmic microwave
background, and that mass increase is itself negligible.

Astrophysical black holes

Black holes in our Universe can be grouped as: primordial black holes,
stellar-mass black holes, and supermassive black holes. The first is
highly speculative; the second and third are broadly accepted.

Primordial black holes of all masses are postulated to have formed
from quantum fluctuations in the early Universe, but those with mass
less than around \(10^{13}\)kg would have already evaporated due to Hawking
radiation. No significant observational evidence has yet been found
of the existence of these objects [MacGibbon and Carr (1991)].

Stellar-mass black holes, ranging from a few to a few tens of solar
masses, are normal but rare endpoints in the evolution of massive
stars. When a star exhausts its nuclear fuel and cools, it must
collapse unless it is supported by nonthermal forces. It is known
that such nonthermal forces cannot resist gravitational
compression for masses greater than around 1.5 to 3\(M_{\rm o}\ .\) (The
uncertainty is due to uncertainty in our knowledge of nuclear physics
at high densities.) There are many stars far more massive than this,
but in their death throes massive stars expel a great deal of their mass in
supernova explosions. Even stars initially as massive as
20-30\(M_{\rm o}\) may blow off enough mass to leave a remnant neutron star
smaller than 1.5\(M_{\rm o}\ .\) Stars more massive than
20-30\(M_{\rm o}\) may
form a neutron star core, only to have it collapse due to fallback of
material from the stellar mantle. Still more massive stars (above
\(\sim40M_{\rm o}\)) may form a black hole directly in collapse, with or
without a supernova explosion. These masses are rather uncertain at
present, as our understanding of supernova explosion mechanisms is
still evolving [Woosley and Janka (2005), Muno (2006)].
Collapsing gas clouds larger than \(\sim100M_{\rm o}\) are expected to
dissipate from radiation pressure, which would prevent more massive
stars from forming, and consequently sets an approximate upper limit
for stellar mass black holes.

Observational evidence for stellar-mass black holes comes primarily
from X-ray astronomy. A black hole in a close binary orbit can form an
accretion disk (see Figure 1) of matter pulled off a normal stellar
companion. In its inspiral, disk material is heated by shearing, and
becomes a strong X-ray emitter. If observations reveal a point-like
X-ray emitter with a mass inferred from orbital dynamics to be above
that possible for a neutron star, it becomes a black hole candidate.
At present there are over a dozen such black hole candidates, including the
first object to be identified as a black hole, the X-ray source
Cygnus X-1 (estimated mass \(10\pm3M_{\rm o}\)) [Casares (2006)].

The other class of observationally-supported black holes is that of
supermassive black holes, ranging from hundreds to billions of solar
masses. Evidence for the existence of black holes at the upper end of
this range is overwhelming. By contrast, the existence of black holes
with mass roughly of order \(10^2\) to \(10^4M_{\rm o}\ ,\) referred to as
"intermediate-mass black holes," is speculative.
The supermassive holes may have begun as
primordial or stellar-mass black holes, but have grown through
absorption of stars or gas, or through mergers with other holes [Ferrarese and Ford (2005)].

Figure 4: Observed orbits of stars near the dark compact object at the center of our Milky Way galaxy: still frame from an animation. Animation created by Prof. Andrea Ghez and her research team at UCLA, from data sets obtained with the W. M. Keck Telescopes: http://www.astro.ucla.edu/~ghezgroup/gc/

Evidence for supermassive black holes originally came from quasars,
small but intense radio sources seen at cosmological distances. Their
huge luminosities implied high mass, while their rapid variability implied
extremely small size. More recently, measurements of Doppler shifts of
stars and gas in the centers of galaxies have shown that compact
objects of mass greater than
\(10^{6}M_{\rm o}\) reside in the cores of most
galaxies; the very small size of these central objects rules out any
plausible alternative to the black hole explanation.
The black
hole at the center of our own Milky Way galaxy has a measured mass of
\(\sim3\times10^6M_{\rm o}\ ,\) based on the orbits of stars near the radio
source Sagittarius A* associated with the Galactic core [Ghez et al. (2005); see Fig. 4].

Gravitational waves from black holes

Black holes are of particular interest for gravitational waves, and
vice versa. For gravitationally-bound systems, the typical maximum
gravitational wave amplitude (dimensionless strain) is \(h\sim
(2GM/c^2)^2/(RD)\sim R_H^2/(RD)\ ,\) where \(D\) is the distance to the
system, \(M\) and \(R\) are the mass and size of the system, and \(R_H\) is
the horizon radius of a black hole of the same mass. From this it is
clear that strongest gravitational waves will involve systems at or
near black hole compactness: in particular, systems containing black
holes in close proximity to one another [Flanagan and Hughes (2005)].

Conversely, a system containing only black holes would not be
expected to radiate any sort of radiation other than
gravitational waves, and in any system containing black holes,
gravitational waves provide the most accurate probes of conditions
deep in their gravitational wells. In particular, gravitational wave
observations could definitively test whether a compact mass is truly a
black hole [Cutler and Thorne (2002)].

The strongest intrinsic source of gravitational waves in the present
Universe is the collision of two comparable-massed black holes. A
sufficiently tight black hole binary system will lose energy through
emission of gravitational waves, causing the orbit to
circularize and then slowly shrink over time.

When the orbital radius approaches the Schwarzschild radius of the
system, complex nonlinear dynamics come into play
and the final stages of the inspiral and merger can only be modeled
through numerical simulations. These simulations are quite
challenging, but recent breakthroughs [Pretorius(2005)(See Fig. 5), Campanelli et al.(2006), Baker et al.(2006)]
have led to the first complete
inspiral-merger gravitational waveforms. After the black holes have
coalesced, they form a single highly-distorted black hole which
quickly settles down to a quiescent Kerr state through emission of
quasinormal "ringdown" gravitational waves
[Pretorius (2005), Campanelli et al. (2006), Baker et al. (2006)].

Another gravitational emission mechanism for supermassive black holes
would come from their capture of compact stellar objects, particularly
stellar-mass black holes. Unlike comparable-mass mergers, these
captures would be on highly eccentric orbits that would not
circularize before merger, resulting in intricate rapidly precessing
orbits in the final months or years of inspiral. The consequent
gravitational-wave signal would allow for exquisitely precise
measurements of the parameters of the supermassive black hole, as well
as tests of any deviation from the predictions of general
relativity [Cutler and Thorne (2002)].

For stellar-mass black holes (tens of \(M_{\rm o}\)), the final stages of
inspiral and merger occur at frequencies of hundreds to thousands of
Hz, a regime targeted by ground-based gravitational-wave
observatories. Systems of supermassive black holes that are millions
of \(M_{\rm o}\) emit at frequencies of tens of mHz, which will be
detectable by proposed space-based gravitational-wave detectors.
Black holes in the billions of \(M_{\rm o}\ ,\) or a stochastic background
of signals from more moderate supermassive black holes at earlier
stages of their inspiral, might emit waves at nHz frequencies that
would be measurable through years-long correlated timing measurements
of radio pulsars [Jenet et al. (2005)].